What is the integral of e^{-sx}*sin(x) ?

The integral of e^{-sx}*sin(x) is -(e^{-sx}cos(x))/(1+s^2)-(se^{-sx}sin(x))/(1+s^2)+C

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integral of e^{-sx}*sin(x)

\int e^{- s x} \cdot sin (x) d x

- \frac{e ^{- s x} cos ( x )}{1 + s ^{2}} - \frac{s e ^{- s x} sin ( x )}{1 + s ^{2}} + C

Solution steps

\int e^{- s x} sin (x) d x

Apply Integration By Parts

: - e^{- s x} cos (x) - \int s e^{- s x} cos (x) d x

= - e^{- s x} cos (x) - \int s e^{- s x} cos (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - e^{- s x} cos (x) - s \cdot \int e^{- s x} cos (x) d x

Apply Integration By Parts

: e^{- s x} sin (x) - \int - s e^{- s x} sin (x) d x

= - e^{- s x} cos (x) - s (e^{- s x} sin (x) - \int - s e^{- s x} sin (x) d x)

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - e^{- s x} cos (x) - s (e^{- s x} sin (x) - (- s \cdot \int e^{- s x} sin (x) d x))

Therefore

\int e^{- s x} sin (x) d x = - e^{- s x} cos (x) - s (e^{- s x} sin (x) - (- s \cdot \int e^{- s x} sin (x) d x))

Isolate

\int e^{- s x} sin (x) d x

= - \frac{e ^{- s x} cos ( x )}{1 + s ^{2}} - \frac{s e ^{- s x} sin ( x )}{1 + s ^{2}}

Add a constant to the solution

= - \frac{e ^{- s x} cos ( x )}{1 + s ^{2}} - \frac{s e ^{- s x} sin ( x )}{1 + s ^{2}} + C

\int (6 sec^{2} (x)) d x

x \to 0 + lim (\frac{9}{x} - \frac{9}{∣ x ∣})

x \to - 2 lim (3 - x)

n = 0 \sum \infty \frac{1 0 ^{n}}{9 ^{n}}

x^{3} + 3 y - x \frac{d y}{d x} = 0

What is the integral of e^{-sx}*sin(x) ?
The integral of e^{-sx}*sin(x) is -(e^{-sx}cos(x))/(1+s^2)-(se^{-sx}sin(x))/(1+s^2)+C